unipotent flows and applicationseskin/lectures-pisa.pdfunipotent flows and applications 5 1.3....

Clay Mathematics ProceedingsVolume 10, 2010

Unipotent Flows and Applications

Alex Eskin

1. General introduction

1.1. Values of indefinite quadratic forms at integral points. The Op-penheim Conjecture. Let

Q(x1, . . . , xn) =∑

1≤i≤j≤n

aijxixj

be a quadratic form in n variables. We always assume that Q is indefinite so that(so that there exists p with 1 ≤ p < n so that after a linear change of variables, Qcan be expresses as:

Q∗p(y1, . . . , yn) =

p∑i=1

y2i −n∑

i=p+1

y2i

We should think of the coefficients aij of Q as real numbers (not necessarilyrational or integer). One can still ask what will happen if one substitutes integersfor the xi. It is easy to see that if Q is a multiple of a form with rational coefficients,then the set of values Q(Zn) is a discrete subset of R. Much deeper is the followingconjecture:

Conjecture 1.1 (Oppenheim, 1929). Suppose Q is not proportional to a ra-tional form and n ≥ 5. Then Q(Zn) is dense in the real line.

This conjecture was extended by Davenport to n ≥ 3.

Theorem 1.2 (Margulis, 1986). The Oppenheim Conjecture is true as long asn ≥ 3. Thus, if n ≥ 3 and Q is not proportional to a rational form, then Q(Zn) isdense in R.

This theorem is a triumph of ergodic theory. Before Margulis, the OppenheimConjecture was attacked by analytic number theory methods. (In particular it wasknown for n ≥ 21, and for diagonal forms with n ≥ 5).

Failure of the Oppenheim Conjecture in dimension 2. Let α > 0 be aquadratic irrational such that α2 ̸∈ Q (e.g. α = (1 +

√5)/2), and let

Q(x1, x2) = x21 − α2x22.

c⃝ 2010 Alex Eskin

1

2 ALEX ESKIN

Proposition 1.3. There exists ϵ > 0 such that for all x1, x2 ∈ Z, |Q(x1, x2)| >ϵ.

Proof. Suppose not. Then for any 1 > ϵ > 0 there exist x1, x2 ∈ Z such that

(1) |Q(x1, x2)| = |x1 − αx2||x1 + αx2| ≤ ϵ.

We may assume x2 ̸= 0. If ϵ < α2, one of the factors must be smaller then α.Without loss of generality, we may assume |x1 − αx2| < α, so |x1 − αx2| < α|x2|.Then,

|x1 + αx2| = |2αx2 + (x1 − αx2)| ≥ 2α|x2| − |x1 − αx2| ≥ α|x2|.

Substituting into (1) we get

(2)

∣∣∣∣x1x2 − α∣∣∣∣ ≤ ϵ|x2||x1 + αx2| ≤ ϵα 1|x2|2 .

But since α is a quadratic irrational, there exists c0 > 0 such that for all p, q ∈ Z,|pq − α| ≥

c0q2 . This is a contradiction to (2) if ϵ < c0α. �

A relation to flows on homogeneous spaces. This was noticed by Raghu-nathan, and previously in implicit form by Cassels and Swinnerton-Dyer. Howeverthe Cassels-Swinnerton-Dyer paper was mostly forgotten. Raghunathan made clearthe connection to unipotent flows, and explained from the point of view of dynamicswhat is different in dimension 2. See §5.1.

1.2. Some basic Ergodic Theory. Transformations, flows and ErgodicMeasures. Let X be a locally compact separable topological space, and T : X →X a map. We assume that there is a finite measure µ on X which is preserved by T .One usually normalizes µ so that µ(X) = 1, in which case µ is called a probabilitymeasure.

Sometimes, instead of a transformation T one considers a flow ϕt, t ∈ R. For afixed t, ϕt is a map from X to X. In this section we state definitions and theoremsfor transformations only, even though we will use them for flows later.

Definition 1.4 (Ergodic Measure). An T -invariant probability measure µ iscalled ergodic for T if for every measurable T -invariant subset E of X one hasµ(E) = 0 or µ(E) = 1.

Every measure can be written as a linear combination (possibly uncountable,dealt with via integration) of ergodic measures. This is called the “ergodic decom-position”.

Ergodic measures always exist. In fact the probability measures form a convexset, and the ergodic probability measures are the extreme points of this set (cf. theKrein-Milman theorem).

Birkhoff’s Ergodic Theorem.

Theorem 1.5 (Birkhoff Ergodic Theorem). Suppose µ is ergodic for T , andsuppose f ∈ L1(X,µ). Then for µ-almost all x ∈ X, we have

(3) limn→∞

1

n

n−1∑k=0

f(Tnx) =

∫X

f dµ.

UNIPOTENT FLOWS AND APPLICATIONS 3

The sum on the left-hand side is called the “time average”, and the integral onthe right is the “space average”. Thus the theorem says that for almost all basepoints x, the time average along the orbit of x converges to the space average.

This theorem is amazing in its generality: the only assumption is ergodicity ofthe measure µ. (This is a some sort of irreducibility assumption).

The set of x ∈ X for which (3) holds is called the generic set for µ.Mutually singular measures. Recall that two probability measures µ1 and µ2are called mutually singular (written as µ1 ⊥ µ2 if there exists a set E such thatµ1(E) = 1, µ2(E) = 0 (so µ2(E

c) = 1).In our proofs we will use repeatedly the following:

Lemma 1.6. Suppose µ1 and µ2 are distinct ergodic measures for the mapT : X → X. Then µ1 ⊥ µ2.

Proof. This is an immediate consequence of the Birkhoff ergodic theorem. Sinceµ1 ̸= µ2 we can find an f such that

∫Xf dµ1 ̸=

∫Xf dµ2. Now let E denote the set

where (3) holds with µ = µ1. �

Remark. It is not difficult to give another proof of Lemma 1.6 using the Radon-Nikodym theorem.

Given an invariant measure µ for T , we want to find conditions under whichit is ivariant under the action of a larger group. Now if H commutes with T , thenfor each h0 ∈ H the measure h0µ is T -invariant. So if µ is ergodic, so is h0µ, andLemma 1.6 applies. More can be said, ([cf. [Ra4, Thm. 2.2], [Mor, Lem. 5.8.6]]):

Lemma 1.7. Suppose T : X → X is preserving an ergodic measure µ. SupposeH is a group with acts continuously on X and commutes with T . Also suppose thatthere exists h0 ∈ H such that h0µ ̸= µ. Then there exists a neighborhood B ofh0 ∈ H and a conull T -invariant subset Ω of X such that

hΩ ∩ Ω = ∅ for all h ∈ B.

Proof. Since h0 commutes with T , the measure h0µ is T -invariant and ergodic.Thus by Lemma 1.6, h0µ ⊥ µ. This implies there is a compact subset K0 of X,such that µ(K0) > 0.99 and K0 ∩h0K0 = ∅. By continuity and compactness, thereare open neighborhoods U and U+ of K0, and a symmetric neighborhood Be of ein H, such that U+ ∩ h0U+ = ∅ and BeU ⊂ U+. From applying (3) with f thecharacteristic function of U , we know there is a conull T -invariant subset Ωh0 of X,such that the T -orbit of every point in Ωh0 spends 99% of its life in U . Now supposethere exists h ∈ Beh0, such that Ωh0 ∩hΩh0 ̸= ∅. Then there exists x ∈ Ωh0 , n ∈ N,and c ∈ Be, such that Tnx and ch0Tnx both belong to U . This implies that Tnxand h0T

nx both belong to U+. This contradicts the fact that U+ ∩ h0U+ = ∅. �

Uniquely ergodic systems. In some applications (in particular to number the-ory) we need some analogue of (3) for all points x (and not almost all). For example,we want to know if Q(Zn) is dense for a specific quadratic form Q (and not for al-most all forms). Then the Birkhoff ergodic theorem is not helpful. However, thereis one situation where we can show that (3) holds for all x.

Definition 1.8. A map T : X → X is called uniquely ergodic if there exists aunique invariant probability measure µ.

4 ALEX ESKIN

Proposition 1.9. Suppose X is compact, T : X → X is uniquely ergodic, andlet µ be the invariant probability measure. Suppose f : X → R is continuous. Thenfor all x ∈ X, (3) holds.

Proof. This is quite easy (as opposed to the Birkhoff ergodic theorem which ishard). Let δn be the probability measure on X defined by

δn(f) =1

n

n−1∑k=0

f(Tnx)

(we are now thinking of measures as elements of the dual space to the space C(X)of continuous functions on X). Note that

δn(f ◦ T ) =1

n

n−1∑k=0

(f ◦ T )(Tnx) = 1n

n∑k=1

f(Tnx),

so

(4) δn(f ◦ T )− δn(f) =1

n(f(x)− f(Tnx)),

(since the sum telescopes). Suppose some subsequence δnj converges to some limitδ∞ (in the weak-* topology). Then, by (4), δ∞(f ◦ T ) = δ∞(f), i.e. δ∞ is T -invariant.

Since X is compact, δ∞ is a probability measure, and thus by the assumptionof unique ergodicity, we have δ∞ = µ. Thus all possible limit points of the sequenceδn are µ. Also the space of probability measures on X is compact (in the weak-*topology), so there exists a convergent subsequence. Hence δn → µ, which is thesame as (3). �

Remarks.

• The main point of the above proof is the construction of an invariantmeasure (namely δ∞) supported on the closure of the orbit of x. Thesame construction works with flows, or more generally with actions ofamenable groups.

• We have used the compactness of X to argue that δ∞ is a probabilitymeasure: this might fail if X is not compact. This phenomenon is called“loss of mass”.

• Of course the problem with Proposition 1.9 is that most of the dynam-ical systems we are interested in are not uniquely ergodic. For exampleany system which has a closed orbit which is not the entire space is notuniquely ergodic.

• However, the proof of Proposition 1.9 suggests that (at least in the amenablecase) the classification of the invariant measures is one of the most power-ful statements one can make about a dynamical system, in the sense thatit allows one to try to understand every orbit (and not just almost everyorbit).

Exercise 1. (To be used in §3.)(a) Show that if α is irrational then the map Tα : [0, 1] → [0, 1] given by

Tα(x) = x+ α (mod 1 ) is uniquely ergodic. Hint: Use Fourier analysis.(b) Use part (a) to show that the flow on R2/Z2 given by ϕt(x, y) = (x +

tα, y + t) is uniquely ergodic.


1.3. Unipotent Flows. Let G be a semisimple Lie group (I will usually as-sume the center of G is finite), and let Γ be a lattice in G (this means that Γ ⊂ Gis a discrete subgroup, and the quotient G/Γ has finite Haar measure). A lattice Γis uniform if G/Γ is compact.

Let U = {ut}t∈R be a unipotent one-parameter subgroup of G. Then U actson G/Γ by left multiplication. (Recall that in SL(n,R) a matrix is unipotent if allits eigenvalues are 1. In a general Lie group an element is unipotent if its Adjoint(acting on the Lie algebra) is a unipotent matrix. ) Examples of unipotent oneparameter subgroups: {(

1 t0 1

), t ∈ R

},

and 1 t t2/20 1 t0 0 1

, t ∈ R ,

Ratner’s measure classification theorem.

Definition 1.10. A probability measure µ on G/Γ is called algebraic if thereexists x̄ ∈ G/Γ and a subgroup F of G such that Fx̄ is closed, and µ is the F -invariant probability measure supported on Fx̄.

Theorem 1.11 (Ratner’s measure classification theorem). Let G be a Lie group,Γ ⊂ G a lattice. Let U be a one-parameter unipotent subgroup of G. Then, anyergodic U -invariant measure is algebraic. (Also the group F in the definition ofalgebraic is generated by unipotent elements, and contains U).

Loosely speaking, this theorem says that all U -invariant ergodic measures arevery nice. The assumption that U is unipotent is crucial: if we consider insteadarbitrary one-parameter subgroups, then there are ergodic invariant measures sup-ported on Cantor sets (and worse). This phenomenon is responsible in particularfor the failure of the Oppenheim conjecture in dimension 2.

Theorem 1.11 has many applications, some of which we will explore in thiscourse. I will give some indication of the ideas which go into the proof of thistheorem in the next two lectures.

Remark on algebraic measures. Let π : G → G/Γ be the projection map.Suppose x̄ ∈ G/Γ, and F ⊂ G is a subgroup. Let StabF (x̄) denote the stabilizer inF of x̄, i.e. the set of elements g ∈ F such that gx̄ = x̄. Then StabF (x̄) = F∩xΓx−1,where x ∈ G is any element such that π(x) = x̄. Thus there is a continuousmap from Fx̄ to F/(F ∩ xΓx−1), which is a bijection, but is in general not ahomeomorphism.

However, in the case of algebraic measures, we are making the additional as-sumption that Fx̄ is closed. In this case, the above map is a homeomorphism, andthus µ is the image under this map of the Haar measure on F/(F ∩ xΓx−1). Theassumption that µ is a probability measure thus implies that F ∩xΓx−1 is a latticein F . (The last condition is usually taken to be part of the definition of an algebraicmeasure).

Uniform Distribution and the classification of orbit closures.

6 ALEX ESKIN

Theorem 1.12 (Ratner’s uniform distribution theorem). Let G be a Lie group,Γ a lattice in G, and U = {ut}t∈R a one-parameter unipotent subgroup. Then forany x̄ ∈ G/Γ there exists a subgroup F ⊃ U (generated by unipotents) with Fx̄closed, and an F -invariant algebraic measure µ supported on Fx̄, such that for anyf ∈ C(G/Γ),

(5) limT→∞

1

T

∫ T0

f(utx̄) dt =

∫Fx̄

f dµ

Remarks.

• It follows from (5) that the closure of the orbit Ux̄ is Fx̄. Thus Theo-rem 1.12 can be rephrased as “any orbit is uniformly distributed in itsclosure”.

• Theorem 1.12 is derived from Theorem 1.11 by an argument morally sim-ilar to the proof of Proposition 1.9. There is one more ingredient: onehas to show that the set of subgroups F which appear in Theorem 1.11is countable up to conjugation (Proposition 4.1 below). For proofs ofthis fact see [Ra6, Theorem 1.1] and [Ra7, Cor. A(2)]), or alternatively[DM4, Proposition 2.1].

An immediate consequence of Theorem 1.12 is the following:

Theorem 1.13 (Raghunathan’s topological conjecture). Let G be a Lie group,Γ ⊂ G a lattice, and U ⊂ G a one-parameter unipotent subgroup. Suppose x̄ ∈ G/Γ.Then there exists a subgroup F of G (generated by unipotents) such that the closureUx̄ of the orbit Ux̄ is Fx̄.

This theorem is due to Ratner in the general case, but several cases were knownpreviously. See §5.1 for a discussion and the relation to the Oppenheim Conjecture.Uniformity of convergence. In many applications it is important to somehowensure that the time averages converge to the space average uniformly in the basepoint x̄ (for example we may have an additional integral over x̄). In the context ofBirkhoff’s ergodic theorem, we have the following:

Lemma 1.14. Suppose ϕt : X → X is a flow preserving an ergodic probabilitymeasure µ. Suppose f ∈ L1(X,µ). Then for any ϵ > 0 and δ > 0, there existsT0 > 0 and a set E ⊂ X with µ(E) < ϵ, such that for any x ∈ Ec and any T > T0we have ∣∣∣∣∣ 1T

∫ T0

f(ϕt(x)) dt−∫X

f dµ

∣∣∣∣∣ < δ(In other words, one has uniform convergence outside of a set of small measure.)

Proof. Let En denote the set of x ∈ X such that for some T > n,∣∣∣∣∣ 1T∫ T0

f(ϕt(x)) dt−∫X

f dµ

∣∣∣∣∣ ≥ δ.Then by the Birkhoff ergodic theorem, µ(

∩∞n=1En) = 0. Hence there exists n ∈ N

such that µ(En) < ϵ. Now let T0 = n, and E = En. �

The uniform distribution theorem of Dani-Margulis. One problem withLemma 1.14 is that it does not provide us with any information about the ex-ceptional set E (other then the fact that it has small measure). In the setting


of unipotent flows, Dani and Margulis proved a theorem (see §4.2 below for theprecise statement) which is the analogue of Lemma 1.14, but with an explicit geo-metric description of the set E. This theorem is crucial for many applications. Itsproof is based on the Ratner measure classification theorem (Theorem 1.11) andthe “linearization” technique of Dani and Margulis (see §4).

2. The case of SL(2,R)/SL(2,Z)

In this lecture I will be loosely following Ratner’s paper [Ra8].

2.1. Basic Preliminaries. The space of lattices. Let G = SL(n,R), andlet Ln denote the space of unimodular lattices in Rn. (By definition, a lattice ∆ isunimodular if an only if the volume of Rn/∆ = 1. ) G acts on Ln as follows: ifg ∈ G and ∆ ∈ Ln is the Z-span of the vectors v1, . . . vn, then gv is the Z-span ofgv1, . . . , gvn. This action is clearly transitive. The stabilizer of the standard latticeZn is Γ = SL(n,Z). This gives an identification of Ln with G/Γ. We choose aright-invariant metric d(·, ·) on G; then this metric descends to G/Γ.

The set Ln(ϵ). For ϵ > 0 let Ln(ϵ) ⊂ Ln denote the set of lattices whose shortestnon-zero vector has length at least ϵ.

Theorem 2.1 (Mahler Compactness). For any ϵ > 0 the set Ln(ϵ) is compact.

The upper half plane. In the rest of this section, we set n = 2. Let K =SO(2) ⊂ G. Given a pair of vectors v1, v2 we can find a unique rotation matrixk ∈ K so that kv1 is pointing along the positive x-axis and kv2 is in the upperhalf plane. The map g =

(v1 v2

)→ kv2 gives an identification of K\G with the

hyperbolic upper half plane H2. Now G (and in particular Γ ⊂ G) acts on K\G bymultiplication on the right. Using the identification of K\G with H2 this becomes(a variant of) the usual action by fractional linear transformations.

The horocycle and geodesic flows. We use the following notation:

ut =

(1 t0 1

)at =

(et 00 e−t

)vt =

(1 0t 1

).

Let U = {ut : t ∈ R}, A = {at : t ∈ R}, V = {vt : t ∈ R}. The action ofU is called the horocycle flow and the action of A is called the geodesic flow. Somebasic commutation relations are the following:

(6) atusa−1t = ue2ts atvsa

−1t = ve−2ts

Thus conjugation by at for t > 0 contracts V and expands U .

Orbits of the geodesic and horocycle flow in the upper half plane. Letp : G → K\G denote the natural projection. Then for x ∈ G, p(Ux) is either ahorizontal line or a circle tangent to the x-axis. Also p(Ax) is either a vertical lineor a semicircular arc orthogonal to the x-axis.

Flowboxes. Let W+ ⊂ U , W− ⊂ V , W0 ⊂ A be intervals containing the identity(we have identified all three subgroups with R). By a flowbox we mean a subset of Gof the form W−W0W+, or one of its right translates by g ∈ G. Clearly, W−W0W+gis an open set containing g. (Recall that in our conventions, right multiplicationby g is an isometry).

8 ALEX ESKIN

2.2. An elementary non-divergence result. Much more is proved in [Kl1].

Lemma 2.2. There exists an absolute constant ϵ0 > 0 such that the followingholds: Suppose ∆ ∈ L2 is a unimodular lattice. Then ∆ cannot contain two linearlyindependent vectors each of length less than ϵ0.

Proof. Let v1 be the shortest vector in ∆, and let v2 be the shortest vector in∆ linearly independent from v1. Then v1 and v2 span a sublattice ∆

′ of ∆. (Infact ∆′ = ∆ but this is not important for us right now). Since ∆ is unimodular,this implies that Vol(R2/∆′) ≥ 1. But Vol(R2/∆′) = ∥v1 × v2∥ ≤ ∥v1∥∥v2∥. Hence∥v1∥∥v2∥ ≥ 1, so the lemma holds with ϵ0 = 1. �

Remark. In general ϵ0 depends on the choice of norm on R2.The following lemma is a simple “nondivergence” result for unipotent orbits:

Lemma 2.3. Suppose ∆ ∈ L2 is a unimodular lattice. Then at least one of thefollowing holds:

(a) ∆ contains a horizontal vector.(b) There exists t ≥ 0 such that a−1t ∆ ∈ L2(ϵ0).

Proof. Suppose ∆ does not contain a horizontal vector, and ∆ ̸∈ L2(ϵ0). Then ∆contains a vector v with ∥v∥ < ϵ0. Since v is not horizontal, there exists a smallestt0 > 0 such that ∥a−1t v∥ = ϵ0. Then by Lemma 2.2 for t ∈ [0, t0], a−1t ∆ contains novectors shorter then ϵ0 (other then a

−1t v and possibly its multiples). In particular

a−1t0 ∆, contains no vectors shorter then ϵ0. This means a−1t0 ∆ ∈ L2(ϵ0). �

Remark. We note that Lemma 2.2 and thus Lemma 2.3 are specific to dimension2.

2.3. The classification of U-invariant measures. Note that for ∆ ∈ L2,the U -orbit of ∆ is closed if and only if ∆ contains a horizontal vector. (Thehorizontal vector is fixed by the action of U). Any closed U -orbit supports a U -invariant probability measure. All such measures are ergodic.

Let ν denote the Haar measure on L2 = G/Γ. The measure ν is normalized sothat ν(L2) = 1. Recall that ν is ergodic for both the horocycle and the geodesicflows (this follows from the Moore ergodicity theorem, see e.g. [BM]).

Our main goal in this lecture is the following:

Theorem 2.4. Suppose µ is an ergodic U -invariant probability measure on L2.Then either µ is supported on a closed orbit, or µ is the Haar measure ν.

Proof. Let L′2 ⊂ L2 denote the set of lattices which contain a horizontal vector.Note that the set L′2 is U -invariant.

Suppose µ is an ergodic U -invariant probability measure on L2. By ergodicityof µ, µ(L′2) = 0 or µ(L′2) = 1. If the latter holds, it is easy to show that µ issupported on a closed orbit. Thus we assume µ(L′2) = 0 and we must show thatµ = ν.

Suppose not. Then there exists a compactly supported continuous functionf : L2 → R and ϵ > 0 such that

(7)

∣∣∣∣∫L2f dµ−

∫L2f dν

∣∣∣∣ > ϵ.


Since f is uniformly continuous, there exists a neighborhoods of the identityW ′0 ⊂ Aand W ′− ⊂ V such that for a ∈W ′0, v ∈W ′− and ∆′′ ∈ L2,

(8) |f(va∆′′)− f(∆′′)| < ϵ/3.

Recall that π : G → G/Γ ∼= L2 denotes the natural projection. Since L2(ϵ0) iscompact the injectivity radius on L2(ϵ0) is bounded from below, hence there existW+ ⊂ U , W0 ⊂ A, W− ⊂ V so that for any g ∈ G with π(g) ∈ L2, the restrictionof π to the flowbox W−W0W+g is injective. We may also assume that W− ⊂ W ′−and W0 ⊂W ′0. Let δ = ν(W−W0W+) denote the Lebesque measure of the flowbox.

By Lemma 1.14 applied to the Lebesque measure ν, there exists a set E ⊂ L2with ν(E) < δ and T1 > 0 such that for any interval I with |I| ≥ T1 and any∆′ ̸∈ E,

(9)

∣∣∣∣ 1|I|∫I

f(ut∆′) dt−

∫L2f dν

∣∣∣∣ < ϵ3 .Now let ∆ be a generic point for U (in the sense of the Birkhoff ergodic theo-

rem). This implies that there exists T2 > 0 such that for any interval I containingthe origin of length greater then T2,

(10)

∣∣∣∣ 1|I|∫I

f(ut∆) dt−∫L2f dµ

∣∣∣∣ < ϵ3 .Since µ(L′2) = 0, we may assume that ∆ does not contain any horizontal vectors.Then by repeatedly applying Lemma 2.3 we can construct arbitrarily large t > 0such that

(11) a−1t ∆ ∈ L2(ϵ).

Now suppose t is such that (11) holds, and consider the set Q = atW−W0W+a−1t ∆.

Then Q can be rewritten as

Q = (atW−a−1t )W0(atW+a

−1t )∆

(so when t is large, Q is long in the U direction and short in A and V directions.)The set Q is an embedded copy of a flowbox in L2, and ν(Q) = δ.

If t is sufficiently large andW−, W0 andW+ are sufficiently small, it is possibleto find for each ∆′ ∈ Q intervals I(∆′) ⊂ R and I(∆) ⊂ R with the followingproperties: |I(∆′)| ≥ max(T1, T2), |I(∆)| ≥ max(T1, T2) and

(12)

∣∣∣∣∣ 1|I(∆′)|∫I(∆′)

f(ut∆′) dt− 1

|I(∆)|

∫I(∆)

f(ut∆) dt

∣∣∣∣∣ < ϵ3 .(this says that the integral of f over a suitably chosen interval of each U -orbit isnearly the same).

Since ν(E) < δ and ν(Q) = δ, there exists ∆′ ∈ Q ∩ Ec. Now (9) holds withI = I(∆′), and (10) holds with I = I(∆). These estimates together with (12)contradict (7). �

Remarks.

• The above proof works with minor modifications if Γ is an arbitrary latticein SL(2,R) (not just SL(2,Z)).

• If Γ is a uniform lattice in SL(2,R) then the horocycle flow on G/Γ isuniquely ergodic. This is a theorem of Furstenberg [F].

10 ALEX ESKIN

• The proof of Theorem 2.4 does not generalize to classification of measuresinvariant under a one-parameter unipotent subgroup on e.g. Ln, n ≥ 3.Completely different ideas are needed. (I will introduce some of them inthe next lecture).

Horospherical subgroups and a theorem of Dani. The key property of Uin dimension 2 which is used in the proof is that U is horospherical, i.e. that it isequal to the set contracted by a one-parameter diagonal subgroup. (One-parameterunipotent subgroups are horospherical only in SL(2,R)). An argument similar inspirit to the proof of Theorem 2.4 can be used to classify the measures invariantunder the action of a horospherical subgroup. This is a theorem of Dani [Dan2](which was proved before Ratner’s measure classification theorem). However, thedetails, and in particular the non-divergence results needed are much more compli-cated.

The horospherical case also allows for an analytic approach, see e.g. [Bu].

3. The case of SL(2,R)nR2.

In this section we will outline a proof of Ratner’s measure classification theoremTheorem 1.11 in the special case G = SL(2,R)nR2, Γ = SL(2,Z)nZ2. We will befollowing the argument of Ratner [Ra1, Ra2, Ra3, Ra4, Ra5, Ra6] and Margulis-Tomanov [MT]. An introduction to these ideas can be found in the books [Mor],and also [BM]. Another exposition of a closely related case is in [EMaMo].

Let X = G/Γ. Then X can be viewed as a space of pairs (∆, v), where ∆is a unimodular lattice in R2 and v is a marked point on the torus R2/∆. (Weremove the translation invariance on the torus R2/∆ since we consider the originas a special point. Alternatively we consider a pair of marked points, and use thetranslation invariance of the torus to place one of the points at the origin). X isthus naturally a fiber bundle where the base is L2 and the fiber above the point∆ ∈ L2 is the torus R2/∆. (X is also sometimes called the universal elliptic curve).

The action of SL(2,R) ⊂ G on X is by left multiplication. It amounts to

g · (∆, v) = (g∆, gv).

The action of the R2 part of G on X is by translating the marked point, i.e forw ∈ R2, w · (∆, v) = (∆, w+ v). Let U be the subgroup of SL(2,R) defined in §2.1.In this lecture our goal is the following special case of Theorem 1.11:

Theorem 3.1. Let µ be an ergodic U -invariant measure on X. Then µ isalgebraic.

Let µ be an ergodic U -invariant measure on X. Let π1 : X → L2 denotethe natural projection (i.e. π1(∆, v) = ∆). Then π

∗1(µ) is an ergodic U -invariant

measure on L2. Thus by Theorem 2.4, either π∗1(µ) is supported on a closed orbitof U , or π∗1(µ) is the Haar measure ν on L2. The first case is easy to handle, so inthe rest of this section we assume that π∗1(µ) = ν. Then we can disintegrate

dµ(∆, v) = dν(∆)dλ∆(v)

where λ∆(v) is some probability measure on the torus R2/∆.


3.1. Finiteness of the fiber measures. Many of the ideas behind the proofof Ratner’s measure classification theorem Theorem 1.11 can be illustrated in theproof of the following:

Proposition 3.2. Either µ is Haar measure on X, or for almost all ∆ ∈ L2,the measure λ∆ is supported on a finite set of points.

We will give an almost complete proof of Proposition 3.2 in this subsection,and then indicate how to complete the proof of Theorem 3.1 in the next subsection.

The subgroups U ,V ,A,H, and W . Let U , V , A be the subgroups of SL(2,R)defined in §2.1. We also give names to certain subgroups of the R2 part of G. Inparticular, let H = {hs, s ∈ R} be the subgroup of G whose action on X is given

by hs(∆, v) = (∆, v + s

(10

)), and W = {wr, r ∈ R} be the subgroup of G whose

action on X is given by wr(∆, v) = (∆, v + r

(01

)). The action of H is called the

horizontal flow and the action of W the vertical flow.

Action of the centralizer. A key observation is that H commutes with U (andso the action of H commutes with the action of U). This implies that if µ isan ergodic U -invariant measure, so is hsµ for any hs ∈ H. (See the discussionpreceeding Lemma 1.7).

Thus, either µ is invariant under H or there exists s ∈ R such that hsµ isdistinct from µ. Suppose µ is invariant under H. Then so are the fiber measuresλ∆ for all ∆ ∈ L2. Then by Exercise 1 (b), for ν-almost all ∆ ∈ L2, λ∆ is theLebesque measure on R2/∆. Thus µ coincides with Haar measure on X for almostall fibers. Then by the ergodicity of µ we can conclude that µ is the Haar measureon X.

Thus, Proposition 3.2 follows from the following:

Proposition 3.3. Suppose µ is not H-invariant. Then for almost all ∆ ∈ L2,the measure λ∆ is supported on a finite set of points.

The element h and the compact set K. From now on, we assume that µ is notH-invariant. Then there exists hs0 ∈ H such that hs0µ ̸= µ. (We may assume thaths0 is fairly close to the identity). Since hs0µ and µ are both ergodic U -invariantmeasures, by Lemma 1.6 we have hs0µ ⊥ µ. Thus the sets of generic points of µ andhs0µ are disjoint. It follows from Lemma 1.7 that there exists δ > 0 and a subsetΩ ⊂ X with µ(Ω) = 1 such that hsΩ∩Ω = ∅ for all s ∈ (s0−δs0, s0]. It follows thatthere exists a compact set K with µ(K) > 0.999 such that for all s ∈ [(1−δ0)s0, s0],hsK ∩K = ∅. Since K is compact and the action of H is continuous, there existϵ > 0 and δ > 0 such that

(13) d(hsK,K) > ϵ for all s ∈ [(1− δ)s0, s0].

The set Ωρ. In view of Lemma 1.14 (with f the characteristic function of K), forany ρ > 0 we can find a set Ωρ with µ(Ωρ) > 1 − ρ and T0 > 0 such that for allT > T0 and all p ∈ Ωρ we have

(14)1

T|{t ∈ [0, T ] : utx ∈ K}| ≥ 1− (0.01)δ

12 ALEX ESKIN

Shearing. Suppose p = (∆, v) and p′ = (∆, v′) are two nearby points in the samefiber. We want to study how they diverge under the action of U . Note that utpand utp

′ are always in the same fiber (i.e. π1(utp) = π1(utp′) = ut∆), but within

the fiber π−11 (ut∆) they will slowly diverge. More precisely, if we let v = (x, y) andv′ = (x′, y′) we have

utv′ − utv = (x′ − x+ t(y′ − y), y′ − y).

Note that if y = y′ (i.e. p and p′ are in the same orbit of H) then utp and utp′ will

not diverge at all.Now suppose y ̸= y′. We are considering the regime where |x′ − x|, |y′ − y|

are very small, but t is so large that d(p, p′) is comparable to 1 (this amounts to|t(y′ − y)| comparable to 1). Under these assumptions, the leading divergence isalong H, i.e.

(15) utp′ = hsutp+ small error

where s = t(y′ − y).

Lemma 3.4. Suppose that for some positive measure set of ∆ ∈ L2, the supportof λ∆ is infinite. Then for any ρ > 0 we can find ∆ ∈ L2 and a sequence of pointspn = (∆, (xn, yn)) ∈ Ωρ which converge to p = (∆, (x, y)) ∈ Ωρ so that yn ̸= y forall n.

We postpone the proof of this lemma (which is intuitively reasonable anyway).

Proof of Proposition 3.3. Suppose the conclusion of Proposition 3.3 is false, sothat for some positive measure set of ∆ ∈ L2, the support of λ∆ is infinite. ThenLemma 3.4 applies.

Let Tn = s0/(yn − y). Then by (15) we have for t ∈ [(1− δ)Tn, Tn],(16) d(utpn, hsutp) < ϵn, where s = t/(y

′ − y).and ϵn → 0 as n → ∞. If n is sufficiently large, then Tn > T0 where T0 is as inthe definition of Ωρ. Then (14) applies to both p and pn, and we can thus findt ∈ [(1 − δ)Tn, Tn] such that utpn ∈ K and also utp ∈ K. Then s = t/(y′ − y) ∈[(1− δ0)s0, s0], and so (16) contradicts (13). �

Proof of Lemma 3.4. Suppose that for some positive measure set of ∆ ∈ L2, thesupport of λ∆ is infinite. Then (by the ergodicity of the action of U on L2), thesupport of λ∆ is infinite for almost all fibers ∆.

Suppose for the moment that the support of λ∆ is countable for almost all∆, so λ∆ is supported on a sequence of points pn with weights λn. But then thecollection of points with the same weight is a U -invariant set, so by ergodicity of µall the points must have the same weight. Thus, since λ∆ is a probability measureif the support of λ∆ is countable it must be finite.

Hence we may assume that the support of λ∆ is uncountable. Then so is Ωρ∩λ∆for almost all ∆. Since any uncountable set contains one of its accumulation points,we may construct a sequence pn ∈ Ωρ with pn → p, where p ∈ Ωρ. It only remainsto verify that if we write pn = (∆, (xn, yn)) and p = (∆, (x, y)) then we can ensureyn ̸= y.

If it is not possible to do so, then it is easy to see that the support of λ∆ iscontained in a finite union of H-orbits. Thus given a < b we can define a functionu((∆, v)) = λ∆({hsv : s ∈ [a, b]}). This function is U -invariant hence constant


for each choice of [a, b]. It is easy to conclude from this that the support of λ∆must be finite. �

3.2. Outline of the Proof of Theorem 3.1. The following general lemmais a stronger version of Lemma 1.14:

Lemma 3.5 (cf. [MT, Lem. 7.3]). Suppose ϕt : X → X is a flow preserving anergodic probability measure µ. For any ρ > 0, there is a “uniformly generic set” Ωρin X, such that

(1) µ(Ωρ) > 1− ρ,(2) for every ϵ > 0 and every compact subset K of X, with µ(K) > 1 − ϵ,

there exists L0 ∈ R+, such that, for all x ∈ Ωρ and all L > L0, we have|{ t ∈ [−L,L] | d(ϕt(x),K) < ϵ } > (1− ϵ)(2L).

Outline of proof. This is similar to that of Lemma 1.14, except that one alsochooses a countable basis of functions and approximates K by elements of thebasis. �

We now return to the setting of §3. Let µ be an ergodic invariant measure forthe action of U on X = G/Γ = (SL(2, R)nR2)/(SL(2,Z)nZ2). For any ρ > 0 wechose a “uniformly generic” set Ωρ for µ as in Lemma 3.5.

The argument of §3.1 is the basis of the following more general proposition(which we state somewhat imprecisely):

Proposition 3.6. Suppose Q is a subgroup of G normalizing U , and supposethat for any ρ > 0 we can find sequences pn and p

′n in Ωρ such that d(pn, p

′n) → 0,

and under the action of U the leading transverse divergence of the trajectories utpnand utp

′n is in the direction of Q (i.e the analogue of (15) holds with q ∈ Q instead

of h ∈ H).Then the measure µ is Q-invariant.

Remark. The analogous statement for unipotent flows is a cornerstone of theproof of Ratner’s Measure Classification Theorem [Ra5, Lem. 3.3], [MT, Lem. 7.5],[Mor, Prop. 5.2.4′].

Remark. For two points in the same fiber, the leading divergence is always alongH (if the points diverge at all). For an arbitrary pair of nearby points in X this isnot the case.

Remark. It is possible that the leading direction of divergence is along U . In thatcase we want to consider the leading “transverse” divergence. In other words wecompare utpn and ut′p

′n where t

′ is chosen to cancel the divergence along U (i.e.one trajectory waits for the other). In that case we say that the leading transversedivergence is along Q if for some q ∈ Q,

utpn = qut′p′n + small error

Remark. To prove Proposition 3.6 we must use Lemma 3.5 instead of Lemma 1.14as in §3.1 because we must choose Ωρ before we know what subgroup Q (and thuswhat compact set K) we will be dealing with.

We now continue the proof of Theorem 3.1. We assume that µ projects to Haarmeasure on L2, but that µ is not Haar measure.

14 ALEX ESKIN

Proposition 3.7. The measure µ is invariant under some subgroup of AHother then H.

Proof. Choose Ωρ as in Lemma 3.5, with ρ = 0.01. By Proposition 3.2, themeasure on each fiber is supported on a finite set. Also we are assuming that µprojects to Haar measure on L2. Then it is easy to see that there exist p ∈ Ωρ,{vn} ⊂ V r {e}, and {wn} ⊂ HW , such that pn = vnwnp ∈ Ωρ, vn → e, andwn → e.

It is not difficult to compute that (after passing to a subsequence), the leadingdirection of divergence of utpn and utp is a one-parameter subgroup Q which iscontained in AH. Then by Proposition 3.6, µ is invariant under Q. By §3.1, wehave Q ̸= H. �

Invariance under A. Any one-parameter subgroup Q of AH other then H isconjugate to A (via an element of H). Thus, by replacing µ with a translateunder H, we may (and will) assume µ is A-invariant.

Note. At this point we do not know that µ is A-ergodic.

Proposition 3.8 (cf. [MT, Cor. 8.4], [Mor, Cor. 5.5.2]). There is a conullsubset Ω of X, such that

Ω ∩ VWp = Ω ∩ V p,for all p ∈ Ω.

Proof. Let Ω be a generic set for for the action of A on X; thus, Ω is conull and,for each p ∈ Ω,

atp ∈ Ωρ for most t ∈ R+.(The existence of such a set follows e.g. from the full version of the Birkhoffergodic theorem, in which one does not assume ergodicity). Given p, p′ ∈ Ω, suchthat p′ = vwp with v ∈ V and w ∈W , we wish to show w = e.

Choose a sequence tn → ∞, such that atnp and atnp′ each belong to Ωρ.Because tn → ∞ and VW is the foliation that is contracted by aR+ , we know thata−tn(vw)atn → e. Furthermore, because A acts on the Lie algebra of V with twicethe weight that it acts on the Lie algebra of W , we see that

∥a−tnvatn∥/|a−tnwatn∥ → 0.Thus p′n = a−tnp

′atn approaches pn = a−tnpatn from the direction of W .If two points p′n and pn approach each other along W , then an easy compu-

tation shows that utpn and utp′n diverge along H. (This observation motivates

Proposition 3.8). Thus by Proposition 3.6 µ must be invariant under H. But thisimpossible by §3.1 (since we are assuming that µ is not Haar measure). �

We require the following entropy estimate, (see [EL] for a proof).

Lemma 3.9 (cf. [MT, Thm. 9.7], [Mor, Prop. 2.5.11]). Suppose W is a closedconnected subgroup of VW that is normalized by a ∈ A+, and let

J(a−1,W) = det((Ad a−1)|LieW

)be the Jacobian of a−1 on W.

(1) If µ is W-invariant, then hµ(a−1) ≥ log J(a−1,W).(2) If there is a conull, Borel subset Ω of X, such that Ω ∩ VWp ⊂ Wp, for

every p ∈ Ω, then hµ(a−1) ≤ log J(a−1,W).


(3) If the hypotheses of 2 are satisfied, and equality holds in its conclusion,then µ is W-invariant.

Proposition 3.10 (cf. [MT, Step 1 of 10.5], [Mor, Prop. 5.6.1]). µ is V -invariant.

Proof. From Lemma 3.9(1), with a−1 in the role of a, we have

log J(a, U) ≤ hµ(a).

From Proposition 3.8 and Lemma 3.9(2), we have

hµ(a−1) ≤ log J(a−1, V ).

Combining these two inequalities with the facts that

• hµ(a) = hµ(a−1) and• J(a, U) = J(a−1, V ),

we have

log J(a, U) ≤ hµ(a) = hµ(a−1) ≤ log J(a−1, V ) = log J(a, U).

Thus, we must have equality throughout, so the desired conclusion follows fromLemma 3.9(3). �

Proposition 3.11. µ is the Lebesgue measure on a single orbit of SL(2,R) onX.

Proof We know:

• U preserves µ (by assumption),• A preserves µ (by Proposition 3.7) and• V preserves µ (by Proposition 3.10).

Since SL(2,R) is generated by U , A and V , µ is SL(2,R) invariant. BecauseSL(2,R) is transitive on the quotient L2 and the support of µ on each fiber is finite(see Proposition 3.2), this implies that some orbit of SL(2,R) has positive measure.By ergodicity of U , then this orbit is conull. �

This completes the proof of Theorem 3.1.

4. Linearization and ergodicity

4.1. Non-ergodic measures invariant under a unipotent. The collec-tion H. (Up to conjugation, this should be the collection of groups which appearin the definition of algebraic measure).

Let G be a Lie group, Γ a discrete subgroup of G, and π : G→ G/Γ the naturalquotient map. Let H be the collection of all closed subgroups F of G such thatF ∩ Γ is a lattice in F and the subgroup generated by unipotent one-parametersubgroups of G contained in F acts ergodically on π(F ) ∼= F/(F ∩ Γ) with respectto the F -invariant probability measure.

Proposition 4.1. The collection H is countable.

16 ALEX ESKIN

Proof. See [Ra6, Theorem 1.1] or [DM4, Proposition 2.1] for different proofs ofthis result. �

Let U be a unipotent one-parameter subgroup of G and F ∈ H. Define

N(F,U) = {g ∈ G : U ⊂ gFg−1}S(F,U) =

∪{N(F ′, U) : F ′ ∈ H, F ′ ⊂ F, dimF ′ < dimF}.

Lemma 4.2. ([MS, Lemma 2.4]) Let g ∈ G and F ∈ H. Then g ∈ N(F,U) \S(F,U) if and only if the group gFg−1 is the smallest closed subgroup of G whichcontains U and whose orbit through π(g) is closed in G/Γ. Moreover in this case theaction of U on gπ(F ) is ergodic with respect to a finite gFg−1-invariant measure.

As a consequence of this lemma,

(17) π(N(F,U) \ S(F,U)) = π(N(F,U)) \ π(S(F,U)), ∀F ∈ H.

Ratner’s theorem [Ra6] states that given any U -ergodic invariant probabilitymeasure on G/Γ, there exists F ∈ H and g ∈ G such that µ is g−1Fg-invariantand µ(π(F )g) = 1. Now decomposing any finite invariant measure into its ergodiccomponent, and using Lemma 4.2, we obtain the following description for any U -invariant probability measure on G/Γ (see [MS, Theorem 2.2]).

Theorem 4.3 (Ratner). Let U be a unipotent one-parameter subgroup of Gand µ be a finite U -invariant measure on G/Γ. For every F ∈ H, let µF denotethe restriction of µ on π(N(F,U) \ S(F,U)). Then µF is U -invariant and any U -ergodic component of µF is a gFg

−1-invariant measure on the closed orbit gπ(F )for some g ∈ N(F,U) \ S(F,U).

In particular, for all Borel measurable subsets A of G/Γ,

µ(A) =∑

F∈H∗µF (A),

where H∗ ⊂ H is a countable set consisting of one representative from each Γ-conjugacy class of elements in H.

Remark. We will often use Theorem 4.3 in the following form: suppose µ is any U -invariant measure on G/Γ which is not Lebesque measure. Then there exists F ∈ Hsuch that µ gives positive measure to some compact subset of N(F,U) \ S(F,U).

4.2. The theorem of Dani-Margulis on uniform convergence. The “lin-earization” technique of Dani and Margulis was devised to understand which mea-sures give positive weight to compact subsets subsets of N(F,U) \ S(F,U). Usingthis technique Dani and Margulis proved the following theorem (which is importantfor many applications, in particular §5):

Theorem 4.4 ([DM4], Theorem 3). Let G be a connected Lie group and let Γbe a lattice in G. Let µ be the G-invariant probability measure on G/Γ. Let U ={ut} be an Ad-unipotent one-parameter subgroup of G and let f be a bounded con-tinuous function on G/Γ. Let D be a compact subset of G/Γ and let ϵ > 0 be given.Then there exist finitely many proper closed subgroups F1 = F1(f,D, ϵ), · · · , Fk =Fk(f,D, ϵ) such that Fi ∩ Γ is a lattice in Fi for all i, and compact subsets C1 =C1(f,D, ϵ), · · · , Ck = Ck(f,D, ϵ) of N(F1, U), · · · , N(Fk, U) respectively, for which


the following holds: For any compact subset K of D −∪

1≤i≤k π(Ci) there exists aT0 ≥ 0 such that for all x ∈ K and T > T0

(18)∣∣∣ 1T

∫ T0

f(utx) dt−∫G/Γ

f dµ∣∣∣ < ϵ.

Remarks.

• This theorem can be informally stated as follows: Fix f and ϵ > 0. Then(18) holds (i.e. the space average of f is within ϵ of the time average of f)uniformly in the base point x, as long as x is restricted to compact setsaway from a finite union of “tubes” N(F,U). (The N(F,U) are associatedwith orbits which do not become equidistributed in G/Γ, because theirclosure is strictly smaller.)

• It is a key point that only finitely many Fk are needed in Theorem 4.4.This has the remarkable implication that if F ∈ H but not one of the Fk,then (18) holds for x ∈ N(F,U) even though Ux is not dense in G/Γ (theclosure of Ux is Fx). Informally, this means the non-dense orbits of Uare themselves becoming equidistributed as they get longer.

A full proof of Theorem 4.4 is beyond the scope of this course. However, wewill describe the “linearization” technique used in its proof in §4.3.

4.3. Ergodicity of limits of ergodic measures. In this subsection we arefollowing [MS], which refers many times to [DM4].

Let P(G/Γ) be the space of all probability measures on G/Γ.

Theorem 4.5 (Mozes-Shah). Let Ui be a sequence of unipotent one-parametersubgroups of G, and for each i, let µi be an ergodic Ui-invariant probability measureon G/Γ. Suppose µi → µ in P(G/Γ). Then there exists a unipotent one-parametersubgroup U such that µ is an ergodic U -invariant measure on G/Γ. In particular,µ is algebraic.

Remarks.

• Let Q(G/Γ) ⊂ P(G/Γ) denote the set of measures ergodic for the actionof a unipotent one-parameter subgroup of G, and let Q0(G/Γ) denoteQ(G/Γ) union the zero measure. If combined with the results of [Kl1,§3], Theorem 4.5 shows that Q0(G/Γ) is compact.

• The theorem actually proved by Mozes and Shah in [MS] gives moreinformation about what kind of limits of ergodic U -invariant measuresare possible. Here is an easily stated consequence:

Suppose xi ∈ G/Γ converge to x∞ ∈ G/Γ, and also xi ∈ Ux∞. Fori ∈ N ∪ {∞} let µi be the algebraic measures supported on Uxi, so thatthe trajectories Uxi are equidistributed with respect to the measures µi.Then µi → µ∞.

We now give some indication of the proof of Theorem 4.5. Let Ui, µi, µ be asin Theorem 4.5. Write Ui = {ui(t)}t∈R.

Invariance of µ under a unipotent.

Lemma 4.6. Suppose Ui ̸= {e} for all large i ∈ N. Then µ is invariant undera one-parameter unipotent subgroup of G.

18 ALEX ESKIN

Proof. For each i ∈ N there exists wi in the Lie algebra g of G, such that∥wi∥ = 1 and Ui = {exp(twi), t ∈ R}. (Here ∥ · ∥ is some Euclidean norm ong). By passing to a subsequence we may assume that wi → w for some w ∈ g,∥w∥ = 1. For any t ∈ R we have Ad(exp(twi)) → Ad(exp(tw)) as i→ ∞. Note thatAd(exp(tw)) is unipotent, since the set of unipotent matrices is closed (consider e.g.the characteristic polynomial). Therefore U = {exp(tw) : t ∈ R} is a nontrivialunipotent subgroup of G. Since exp twi → exp tw for all t and µi → µ, it followsthat µ is invariant under the action of U on G/Γ. �

Application of Ratner’s measure classification theorem. We want to ana-lyze the case when the limit measure µ is not the G-invariant measure. By Ratner’sdescription of µ as in Theorem 4.3, there exists a proper subgroup F ∈ H, ϵ0 > 0,and a compact set C1 ⊂ N(F,U) \ S(F,U) such that µ(π(C1)) > ϵ0. Thus forany neighborhood Φ of π(C1), we have µi(Φ) > ϵ0 for all large i ∈ N. Thus theunipotent trajectories which are equidistributed with respect to the measures µispend a fixed proportion of time in Φ.

Linearization of neighborhoods of singular subsets. Let F ∈ H. Let gdenote the Lie algebra of G and let f denote its Lie subalgebra associated to F .For d = dim f, put VF = ∧df, the d-th exterior power, and consider the linear G-action on VF via the representation ∧d Ad, the d-th exterior power of the Adjointrepresentation of G on g. Fix pF ∈ ∧df \ {0}, and let ηF : G → VF be the mapdefined by ηF (g) = g · pF = (∧d Ad g) · pF for all g ∈ G. Note that

ηF−1(pF ) = {g ∈ NG(F ) : det(Ad g|f) = 1}.

Remark. The idea of Dani and Margulis is to work in the representation spaceVF (or more precisely V̄F , which is the quotient of VF by the involution v → −v)instead of G/Γ. In fact, for most of the argument one works only with the oribitG ·pF ⊂ VF . The advantage is that F is collapsed to a point (since it stabilizes pF ).The difficulty is that the map ηF : G → V̄F is not Γ-equivariant, and so becomesmultivalued if considered as a map from G/Γ to VF .

Proposition 4.7 ([DM4, Theorem 3.4]). The orbit Γ · pF is discrete in VF .

Remark. In the arithmetic case the above proposition is immediate.

Proposition 4.8. ([DM4, Prop. 3.2]) Let AF be the linear span of ηF (N(F,U))in VF . Then

ηF−1(AF ) = N(F,U).

Let NG(F ) denote the normalizer in G of F . Put ΓF = NG(F ) ∩ Γ. Then forany γ ∈ ΓF , we have γπ(F ) = π(F ), and hence γ preserves the volume of π(F ).Therefore |det(Ad γ|f)| = 1. Hence γ · pF = ±pF . Now define

V̄F =

{VF /{Id,-Id} if ΓF · pF = {pF ,−pF }VF if ΓF · pF = pF

The action of G factors through the quotient map of VF onto V̄F . Let p̄F denotethe image of pF in V̄F , and define η̄F : G → V̄F as η̄F (g) = g · p̄F for all g ∈ G.Then ΓF = η̄F

−1(p̄F ) ∩ Γ. Let ĀF denote the image of AF in V̄F . Note that theinverse image of ĀF in VF is AF .


For every x ∈ G/Γ, define the set of representatives of x in V̄F to be

Rep(x) = η̄F (π−1(x)) = η̄F (xΓ) ⊂ V̄F .

Remark. If one attempts to consider the map η̄F : G→ V̄F as a map from G/Γ toV̄F , one obtains the multivalued map which takes x ∈ G/Γ to the set Rep(x) ⊂ V̄F .

The following lemma allows us to understand the map Rep in a special case:

Lemma 4.9. If x = π(g) and g ∈ N(F,U) \ S(F,U)

Rep(x) ∩ ĀF = {g · pF }.

Thus x has a single representative in ĀF ⊂ VF .

Proof. Indeed, using Proposition 4.8,

Rep(π(g)) ∩ ĀF = (gΓ ∩N(F,U)) · p̄FNow suppose γ ∈ Γ is such that gγ ∈ N(F,U). Then g belongs to N(γFγ−1, U) aswell as N(F,U). Since g ̸∈ S(F,U), we must have γFγ−1 = F , so γ ∈ ΓF . Thenγp̄F = p̄F , so (gΓ ∩N(F,U)) · p̄F = {g · p̄F } as required. �

We extend this observation in the following result (cf. [Sha1, Prop. 6.5]).

Proposition 4.10 ([DM4, Corollary 3.5]). Let D be a compact subset of ĀF .Then for any compact set K ⊂ G/Γ \ π(S(F,U)), there exists a neighborhood Φ ofD in V̄F such that any x ∈ K has at most one representative in Φ.

Remark. This proposition constructs a “fundamental domain” Φ around anycompact subset D of ĀF , so that for any x in a compact subset of G/Γ away fromπ(S(F,U)), Rep(x) has at most one element in Φ. Using this proposition, one canuniquely represent in Φ the parts of the unipotent trajectories in G/Γ lying in K.

Proposition 4.11 ([DM4, Proposition 4.2]). Let a compact set C ⊂ ĀF andan ϵ > 0 be given. Then there exists a (larger) compact set D ⊂ ĀF with thefollowing property: For any neighborhood Φ of D in V̄F there exists a neighborhoodΨ of C in V̄F with Ψ ⊂ Φ such that the following holds: For any unipotent oneparameter subgroup {u(t)} of G, an element w ∈ V̄H and and interval I ⊂ R, ifu(t0)w ̸∈ Φ for some t0 ∈ I then,

(19) |{t ∈ I : u(t)w ∈ Ψ}| ≤ ϵ · |{t ∈ I : u(t)w ∈ Φ}|.

Proof. This is a “polynomial divergence” estimate similar to these in [Kl1, §2]and [Kl1, §3] �

Proposition 4.12. Let ϵ > 0, a compact set K ⊂ G/Γ \ π(S(F,U)), and acompact set C ⊂ ĀF be given. Then there exists a neighborhood Ψ of C in V̄F suchthat for any unipotent one-parameter subgroup {u(t)} of G and any x ∈ G/Γ, atleast one of the following conditions is satisfied:

(1) There exists w ∈ Rep(x) ∩ Ψ such that {u(t)} ⊂ Gw, where Gw = {g ∈G : gw = w}.

(2) For all large T > 0,

|{t ∈ [0, T ] : u(t)x ∈ K ∩ π(η̄−1F (Ψ))}| ≤ ϵT.

20 ALEX ESKIN

ΦΨ

CD

ĀF

ĀF

u(t)w

Figure 1. Proposition 4.11.

Proof. Let a compact set D ⊂ ĀF be as in Proposition 4.11. Let Φ be a givenneighborhood of D in V̄F . Replacing Φ by a smaller neighborhood of D, by Propo-sition 4.10 the set Rep(x) ∩ Φ contains at most one element for all x ∈ K. By thechoice of D there exists a neighborhood Ψ of C contained in Φ such that equa-tion (19) holds.

Now put Ω = π(η̄−1F (Ψ)) ∩K, and define

(20) E = {t ≥ 0 : u(t)x ∈ Ω}.Let t ∈ E. By the choice of Φ, there exists a unique w ∈ V̄F such that Rep(u(t)x)∩Φ = {u(t)w}.

Since s → u(s)w is a polynomial function, either it is constant or it is un-bounded as s → ±∞. In the first case condition 1) is satisfied and we are done.Now suppose that condition 1 does not hold. Then for every t ∈ E, there exists alargest open interval I(t) ⊂ (0, T ) containing t such that(21) u(s)w ∈ Φ for all s ∈ I(t).Put I = {I(t) : t ∈ E}, Then for any I1 ∈ I and s ∈ I1 ∩ E, we have I(s) = I1.Therefore for any t1, t2 ∈ E, if t1 < t2 then either I(t1) = I(t2) or I(t1) ∩ I(t2) ⊂(t1, t2). Hence any t ∈ [0, T ] is contained in at most two distinct elements of I.Thus

(22)∑I∈I

|I| ≤ 2T.

Now by equations (19) and (21), for any t ∈ E,(23) |{s ∈ I(t) : u(s)w ∈ Ψ}| < ϵ · |I(t)|.Therefore by equations (22) and (23), we get

|E| ≤ ϵ ·∑I∈I

|I| ≤ (2ϵ)T,

which is condition 2 for 2ϵ in place of ϵ. �


Outline of Proof of Theorem 4.5. Suppose µ is not Haar measure on G/Γ. ByLemma 4.6 µ is invariant under some one-parameter unipotent subgroup µ. Thenby Theorem 4.3 there exists F ∈ H such that µ(N(F,U)) > 0 and µ(S(F,U)) = 0.Thus there exists a compact subset C1 of N(F,U) \ S(F,U) and α > 0 such that

(24) µ(π(C1)) > α.

Take any y ∈ π(C1). It is easy to see that for each i ∈ N there exists yi ∈supp(µi) such that {ui(t)yi} is uniformly distributed with respect to µi, and alsoyi → y as i→ ∞. Let hi → e be a sequence in G such that hiyi = y for all i ∈ N.

We now replace µi by µ′i = hiµi. We still have µ

′i → µ, but now we also have

y ∈ supp(µ′i) for all i. Let u′i(t) = hiui(t)h−1i . Then the trajectory {u′i(t)y} is

uniformly distributed with respect to µ′i.We now apply Proposition 4.12 for C = η̄F (C1) and ϵ = α/2. We can choose a

compact neighborhoodK of π(C1) such thatK∩S(F,U) = ∅. Put Ω = π(η̄−1F (Ψ))∩K. Since µ′i → µ, due to (24) there exists k0 ∈ N such that µ′i(Ω) > ϵ for all i ≥ k0.This means that Condition 2) of Proposition 4.12 is violated for all i ≥ k0. Thereforeaccording to condition 1) of Proposition 4.12, for each i ≥ k0,

{u′i(t)y}t∈R ⊂ Gwy,

where Gw is as in Proposition 4.12. By Proposition 4.7, Gwy is closed in G/Γ.The rest of the proof is by induction on dimG. If dimGw < dimG then

everything is taking place in the homogeneous space Gwy, and therefore µ is ergodicby the induction hypothesis. If dimGw = dimG then Gw = G and hence Fis a normal subgroup of G. In this case one can project the measures to thehomogeneous space G/(FΓ) and apply induction. �

5. Oppenheim and Quantitative Oppenheim

5.1. The Oppenheim Conjecture. Let Q be an indefinite nondegeneratequadratic form in n variables. Let Q(Zn) denote the set of values of Q at integralpoints. The Oppenheim conjecture, proved by Margulis (cf. [Mar3]) states that ifn ≥ 3, and Q is not proportional to a form with rational coefficients, then Q(Zn)is dense. The Oppenheim conjecture enjoyed attention and many studies since itwas conjectured in 1929 mostly using analytic number theory methods.

In the mid seventies Raghunathan observed a remarkable connection betweenthe Oppenheim Conjecture and unipotent flows on the space of lattices Ln =SL(n,R)/SL(n,Z). It can be summarized as the following:

Observation 5.1 (Raghunathan). Let Q be an indefinite quadratic form Qand let H = SO(Q) denote its orthogonal group. Consider the orbit of the standardlattice Zn ∈ Ln under H. Then the following are equivalent:

(a) The orbit HZn is not relatively compact in Ln.(b) For all ϵ > 0 there exists u ∈ Zn such that 0 < |Q(u)| < ϵ.(c) The set Q(Zn) is dense in R.

Proof. Suppose (a) holds, so some sequence hkZn leaves all compact sets. Then inview of the Mahler compactness criterion there exist vk ∈ hkZn such that ∥vk∥ → 0.Then also by continuity, Q(vk) → 0. But then h−1k vk ∈ Zn, and Q(h

−1k vk) =

Q(vk) → 0. Thus (b) holds.

22 ALEX ESKIN

It is easy to see that (b) implies (a). It is also possible to show that (b) implies(c). �

The Oppenheim Conjecture, the Raghunathan Conjecture and Unipo-tent Flows. Raghunathan also explained why the case n = 2 is different: inthat case H = SO(Q) is not generated by unipotent elements. Margulis’s proof ofthe Oppenheim conjecture, given in [Mar 2-4] uses Raghunathan’s observation.In fact Margulis showed that that any relatively compact orbit of SO(2, 1) inSL(3,R)/SL(3,Z) is compact; this implies the Oppenheim Conjecture.

Raghunathan also conjectured Theorem 1.13. In the literature it was first statedin the paper [Dan2] and in a more general form in [Mar3] (when the subgroup Uis not necessarily unipotent but generated by unipotent elements). Raghunathan’sconjecture was eventually proved in full generality by M. Ratner (see [Ra7]). Earlierit was known in the following cases: (a) G is reductive and U is horospherical (see[Dan2]); (b) G = SL(3,R) and U = {u(t)} is a one-parameter unipotent subgroupof G such that u(t)− I has rank 2 for all t ̸= 0, where I is the identity matrix (see[DM2]); (c) G is solvable (see [Sta1] and [Sta2]). We remark that the proof givenin [Dan2] is restricted to horospherical U and the proof given in [Sta1] and [Sta2]cannot be applied for nonsolvable G.

However the proof in [DM2] together with the methods developed in [Mar 2-4]and [DM1] suggest an approach for proving the Raghunathan conjecture in generalby studying the minimal invariant sets, and the limits of orbits of sequences of pointstending to a minimal invariant set. This strategy can be outlined as follows: Letx be a point in G/Γ, and U a connected unipotent subgroup of G. Denote byX the closure of Ux and consider a minimal closed U -invariant subset Y of X.Suppose that Ux is not closed (equivalently X is not equal to Ux). Then X shouldcontain ”many” translations of Y by elements from the normalizer N(U) of U notbelonging to U . After that one can try to prove that X contains orbits of biggerand bigger unipotent subgroups until one reaches horospherical subgroups. Thebasic tool in this strategy is the following fact. Let y be a point in X, and let gnbe a sequence of elements in G such that gn converges to 1, gn does not belong toN(U), and yn = gny belongs to X. Then X contains AY where A is a nontrivialconnected subset in N(U) containing 1 and ”transversal” to U . To prove this onehas to observe that the orbits Uyn and Uy are ”almost parallel” in the direction ofN(U) most of the time in ”the intermediate range”. (cf. Proposition 3.6).

In fact the set AU as a subset of N(U)/U is the image of a nontrivial rationalmap from U into N(U)/U . Moreover this rational map sends 1 to 1 and alsocomes from a polynomial map from U into the closure of G/U in the affine space Vcontaining G/U . This affine space V is the space of the rational representation ofG such that V contains a vector the stabilizer of which is U (Chevalley theorem).

This program was being actively pursued at the time Ratner’s results wereannounced (cf. [Sha3]).

5.2. A quantitative version of the Oppenheim Conjecture. Referencesfor this subsection are [EMM1] and [EMM2].

In this section we study some finer questions related to the distribution of thevalues of Q at integral points.

Let ν be a continuous positive function on the sphere {v ∈ Rn | ∥v∥ = 1}, andlet Ω = {v ∈ Rn | ∥v∥ < ν(v/∥v∥)}. We denote by TΩ the dilate of Ω by T . Define


the following set:

V Q(a,b)(R) = {x ∈ Rn | a < Q(x) < b}

We shall use V(a,b) = VQ(a,b) when there is no confusion about the form Q. Also

let V(a,b)(Z) = V Q(a,b)(Z) = {x ∈ Zn | a < Q(x) < b}. The set TΩ ∩ Zn consists

of O(Tn) points, Q(TΩ ∩ Zn) is contained in an interval of the form [−µT 2, µT 2],where µ > 0 is a constant depending on Q and Ω. Thus one might expect that forany interval [a, b], as T → ∞,(25) |V(a,b)(Z) ∩ TΩ| ∼ cQ,Ω(b− a)Tn−2

where cQ,Ω is a constant depending on Q and Ω. This may be interpreted as“uniform distribution” of the sets Q(Zn ∩ TΩ) in the real line. The main result ofthis section is that (25) holds if Q is not proportional to a rational form, and hassignature (p, q) with p ≥ 3, q ≥ 1. We also determine the constant cQ,Ω.

If Q is an indefinite quadratic form in n variables, Ω is as above and (a, b) isan interval, we show that there exists a constant λ = λQ,Ω so that as T → ∞,(26) Vol(V(a,b)(R) ∩ TΩ) ∼ λQ,Ω(b− a)Tn−2

The main result is the following:

Theorem 5.2. Let Q be an indefinite quadratic form of signature (p, q), withp ≥ 3 and q ≥ 1. Suppose Q is not proportional to a rational form. Then for anyinterval (a, b), as T → ∞,(27) |V(a,b)(Z) ∩ TΩ| ∼ λQ,Ω(b− a)Tn−2

where n = p+ q, and λQ,Ω is as in (26).

The asymptotically exact lower bound was proved in [DM4]. Also a lowerbound with a smaller constant was obtained independently by M. Ratner, and byS. G. Dani jointly with S. Mozes (both unpublished). The upper bound was provedin [EMM1].

If the signature of Q is (2, 1) or (2, 2) then no universal formula like (25) holds.In fact, we have the following theorem:

Theorem 5.3. Let Ω0 be the unit ball, and let q = 1 or 2. Then for everyϵ > 0 and every interval (a, b) there exists a quadratic form Q of signature (2, q)not proportional to a rational form, and a constant c > 0 such that for an infinitesequence Tj → ∞,

|V(a,b)(Z) ∩ TΩ0| > cT qj (log Tj)1−ϵ.

The case q = 1, b ≤ 0 of Theorem 5.3 was noticed by P. Sarnak and worked outin detail in [Bre]. The quadratic forms constructed are of the form x21 + x

22 − αx23,

or x21 + x22 − α(x23 + x24), where α is extremely well approximated by squares of

rational numbers.However in the (2, 1) and (2, 2) cases, one can still establish an upper bound

of the form cT q log T . This upper bound is effective, and is uniform over compactsets in the set of quadratic forms. We also give an effective uniform upper boundfor the case p ≥ 3.

Theorem 5.4 ([EMM1]). Let O(p, q) denote the space of quadratic forms ofsignature (p, q) and discriminant ±1, let n = p + q, (a, b) be an interval, and letD be a compact subset of O(p, q). Let ν be a continuous positive function on the

24 ALEX ESKIN

unit sphere and let Ω = {v ∈ Rn | ∥v∥ < ν(v/∥v∥)}. Then, if p ≥ 3 there existsa constant c depending only on D, (a, b) and Ω such that for any Q ∈ D and allT > 1,

|V(a,b)(Z) ∩ TΩ| < cTn−2

If p = 2 and q = 1 or q = 2, then there exists a constant c > 0 depending only onD, (a, b) and Ω such that for any Q ∈ D and all T > 2,

|V(a,b) ∩ TΩ ∩ Zn| < cTn−2 log T

Also, for the (2, 1) and (2, 2) cases, we have the following “almost everywhere”result:

Theorem 5.5. For almost all quadratic forms Q of signature (p, q) = (2, 1) or(2, 2)

|V(a,b)(Z) ∩ TΩ| ∼ λQ,Ω(b− a)Tn−2

where n = p+ q, and λQ,Ω is as in (26).

Theorem 5.5 may be proved using a recent general result of Nevo and Stein[NS]; see also [EMM1].

It is also possible to give a “uniform” version of Theorem 5.2, following [DM4]:

Theorem 5.6. Let D be a compact subset of O(p, q), with p ≥ 3. Let n = p+q,and let Ω be as in Theorem 5.4. Then for every interval [a, b] and every θ > 0,there exists a finite subset P of D such that each Q ∈ P is a scalar multiple of arational form and for any compact subset F of D −P there exists T0 such that forall Q in F and T ≥ T0,

(1− θ)λQ,Ω(b− a)Tn−2 ≤ |V(a,b)(Z) ∩ TΩ| ≤ (1 + θ)λQ,Ω(b− a)Tn−2

where λQ,Ω is as in (26).

As in Theorem 5.2 the upper bound is from [EMM1]; the asymptotically exactlower bound, which holds even for SO(2, 1) and SO(2, 2), was proved in [DM4].

Remark 5.7. If we consider |V(a,b)(R)∩TΩ∩P(Zn)| instead of |V(a,b)(Z)∩TΩ|(where P(Zn) denotes the set of primitive lattice points, then Theorem 5.2 andTheorem 5.6 hold provided one replaces λQ,Ω by λ

′Q,Ω = λQ,Ω/ζ(n), where ζ is the

Riemann zeta function.

More on signature (2,2). Recall that a subspace is called isotropic if the re-striction of the quadratic form to the subspace is identically zero. Observe alsothat whenever a form of signature (2, 2) has a rational isotropic subspace L thenL ∩ TΩ contains on the order of T 2 integral points x for which Q(x) = 0, henceNQ,Ω(−ϵ, ϵ, T ) ≥ cT 2, independently of the choice of ϵ. Thus to obtain an as-ymptotic formula similar to (27) in the signature (2, 2) case, we must exclude thecontribution of the rational isotropic subspaces. We remark that an irrational qua-dratic form of signature (2, 2) may have at most 4 rational isotropic subspaces (see[EMM2, Lemma 10.3]).

The space of quadratic forms in 4 variables is a linear space of dimension 10.Fix a norm ∥ · ∥ on this space.


Definition 5.8. (EWAS) A quadratic form Q is called extremely well approx-imable by split forms (EWAS) if for any N > 0 there exists a split integral form Q′

and 2 ≤ k ∈ R such that ∥∥∥∥Q− 1kQ′∥∥∥∥ ≤ 1kN .

The main result of [EMM2] is:

Theorem 5.9. Suppose Ω is as above. Let Q be an indefinite quadratic formof signature (2, 2) which is not EWAS. Then for any interval (a, b), as T → ∞,

(28) ÑQ,Ω(a, b, T ) ∼ λQ,Ω(b− a)T 2,

where the constant λQ,Ω is as in (26), and ÑQ,Ω counts the points not contained inisotropic subspaces.

Open Problem. State and prove a result similar to Theorem 5.9 for the signature(2, 1) case.

Eigenvalue spacings on flat 2-tori. It has been suggested by Berry and Taborthat the eigenvalues of the quantization of a completely integrable Hamiltonianfollow the statistics of a Poisson point-process, which means their consecutive spac-ings should be i.i.d. exponentially distributed. For the Hamiltonian which is thegeodesic flow on the flat 2-torus, it was noted by P. Sarnak [Sar] that this problemtranslates to one of the spacing between the values at integers of a binary quadraticform, and is related to the quantitative Oppenheim problem in the signature (2, 2)case. We briefly recall the connection following [Sar].

Let ∆ ⊂ R2 be a lattice and let M = R2/∆ denote the associated flat torus.The eigenfunctions of the Laplacian on M are of the form fv(·) = e2πi⟨v,·⟩, where vbelongs to the dual lattice ∆∗. The corresponding eigenvalues are 4π2∥v∥2, v ∈ ∆∗.These are the values at integral points of the binary quadratic B(m,n) = 4π2∥mv1+nv2∥2, where {v1, v2} is a Z-basis for ∆∗. We will identify ∆∗ with Z2 using thisbasis.

We label the eigenvalues (with multiplicity) by

0 = λ0(M) < λ1(M) ≤ λ2(M) . . .It is easy to see that Weyl’s law holds, i.e.

|{j : λj(M) ≤ T}| ∼ cMT,where cM = (areaM)/(4π). We are interested in the distribution of the localspacings λj(M)− λk(M). In particular, for 0 ̸∈ (a, b), set

RM (a, b, T ) =|{(j, k) : λj(M) ≤ T, λk(M) ≤ T, a ≤ λj(M)− λk(M) ≤ b}|

T.

The statistic RM is called the pair correlation. The Poisson-random model predicts,in particular, that

(29) limT→∞

RM (a, b, T ) = c2M (b− a).

Note that the differences λj(M) − λk(M) are precisely the integral values of thequadratic form QM (x1, x2, x3, x4) = B(x1, x2)−B(x3, x4).

P. Sarnak showed in [Sar] that (29) holds on a set of full measure in the spaceof tori. Some remarkable related results for forms of higher degree and higherdimensional tori were proved in [V1], [V2] and [V3]. These methods, however,

26 ALEX ESKIN

cannot be used to explicitly construct a specific torus for which (29) holds. Acorollary of Theorem 5.9 is the following:

Theorem 5.10. Let M be a 2 dimensional flat torus rescaled so that one ofthe coefficients in the associated binary quadratic form B is 1. Let A1, A2 denotethe two other coefficients of B. Suppose that there exists N > 0 such that for alltriples of integers (p1, p2, q) with q ≥ 2,

maxi=1,2

∣∣∣∣Ai − piq∣∣∣∣ > 1qN .

Then, for any interval (a, b) not containing 0, (29) holds, i.e.

limT→∞

RM (a, b, T ) = c2M (b− a).

In particular, the set of (A1, A2) ⊂ R2 for which (29) does not hold has zero Haus-dorff dimension.

Thus, if one of the Ai is Diophantine’s (e.g. algebraic), then M has a spectrumwhose pair correlation satisfies the Berry-Tabor conjecture.

This establishes the pair correlation for the flat torus or “boxed oscillator” con-sidered numerically by Berry and Tabor. We note that without some diophantinecondition, (29) may fail.

5.3. Passage to the space of lattices. We now relate the counting problemof Theorem 5.2 to a certain integral expression involving the orthogonal group ofthe quadratic form and the space of lattices SL(n,R)/SL(n,Z). Roughly this isdone as follows. Let f be a bounded function on Rn − {0} vanishing outside acompact subset. For a lattice ∆ ∈ Ln let

(30) f̃(∆) =∑

v∈∆\{0}

f(∆)

(the function f̃ is called the “Siegel Transform” of f). The proof is based on theidentity of the form

(31)

∫K

f̃(atk∆) dk =∑

v∈∆\{0}

∫K

f(atkv) dk

obtained by integrating (30). In (31) {at} is a certain diagonal subgroup of theorthogonal group of Q, and K is a maximal compact subgroup of the orthogonalgroup of Q. Then for an appropriate function f , the right hand side is then relatedto the number of lattice points v ∈ [et/2, et]∂Ω with a < Q(v) < b. The asymptoticsof the left-hand side is then established using the ergodic theory of unipotent flowsand some other techniques.

Quadratic Forms, and the lattice ∆Q. Let n ≥ 3, and let p ≥ 2. We denoten− p by q, and assume q > 0. Let {e1, e2, . . . en} be the standard basis of Rn. LetQ0 be the quadratic form defined by

(32) Q0

(n∑

i=1

viei

)= 2v1vn +

p∑i=2

v2i −n−1∑

i=p+1

v2i for all v1, . . . , vn ∈ R.

It is straightforward to verify that Q0 has signature (p, q). Let G = SL(n,R), thegroup of n × n matrices of determinant 1. For each quadratic form Q and g ∈ G,


let Qg denote the quadratic form defined by Qg(v) = Q(gv) for all v ∈ Rn. Bythe well known classification of quadratic forms over R, for each Q ∈ O(p, q) thereexists g ∈ G such that Q = Qg0. Then let ∆Q denote the lattice gZn, so thatQ0(∆Q) = Q(Zn).

For any quadratic form Q let SO(Q) denote the special orthogonal group cor-responding to Q; namely {g ∈ G | Qg = Q}. Let H = SO(Q0). Then the mapH\G→ O(p, q) given by Hg → Qg0 is a homeomorphism.

The map at and the group K. For t ∈ R, let at be the linear map so thatate1 = e

−te1, aten = eten, and atei = ei, 2 ≤ i ≤ n − 1. Then the one-parameter

group {at} is contained in H. Let K̂ be the subgroup of G consisting of orthogonalmatrices, and let K = H ∩ K̂. It is easy to check that K is a maximal compactsubgroup of H, and consists of all h ∈ H leaving invariant the subspace spannedby {e1 + en, e2, . . . , ep}. We denote by m the normalized Haar measure on K.

A Lemma about vectors in Rn. In this section we will be somewhat informal.For a completely rigorous argument see [EMM1, §§3.4-3.5]. Also for simplicity welet ν = 1 in this section.

LetW ⊂ Rn be the characteristic function of the region defined by the inequal-ities on x = (x1, . . . , xn):

a ≤ Q0(x) ≤ b, (1/2) ≤ ∥x∥ ≤ 2,x1 > 0, (1/2)x1 ≤ |xi| ≤ (1/2)x1 for 2 ≤ i ≤ n− 1.

Let f be the characteristic function of W .

Lemma 5.11. There exists T0 > 0 such that for every t with et > T0, and every

v ∈ Rn with ∥v∥ > T0,

(33) cp,qe(n−2)t

∫K

f(atkv) dm(k) ≈

{1 if a ≤ Q0(x) ≤ b and e

t

2 ≤ ∥v∥ ≤ et,

0 otherwise

where cp,q is a constant depending only on p and q.

Proof. This is a direct calculation. �

Remark. The ≈ in (33) is essentially equality up to “edge effects”. These edgeeffects can be overcome if one approximated f from above and below by continuousfunctions f+ and f− in such a way that the L

1 norm of f+−f− is small. We choosenot to do this here in order to not clutter the notation.

In (33), we let T = et and sum over v ∈ ∆Q. We obtain:

Proposition 5.12. As T → ∞,

cp,qTn−2

∫K

f̃(atk∆Q) ≈ |{v ∈ ∆Q : a < Q0(v) < b and 12T ≤ ∥v∥ ≤ T}|,

where t = log T . Note that the right-hand side is by definition |V Q(a,b)(Z)∩[T/2, T ]Ω0|,where Ω0 is the unit ball.

We also note without proof the following lemma:

28 ALEX ESKIN

Lemma 5.13. Let ρ be a continuous positive function on the sphere, and letΩ = {v ∈ Rn |∥v∥ < ρ(v/∥v∥)}. Then there exists a constant λ = λQ,Ω so that asT → ∞,

Vol(V Q(a,b)(R) ∩ TΩ) ∼ λQ,Ω(b− a)Tn−2.

Also (using Siegel’s formula), cp,q∫Ln f̃ = cp,q

∫Rn f = (1− 2

2−n)λQ,Ω.

Remark. One can verify that:

λQ,Ω =

∫L∩Ω

dA

∥∇Q∥,

where L is the lightcone Q = 0 and dA is the area element on L.

The main theorems. In view of Proposition 5.12 and Lemma 5.13, to proveTheorem 5.2 one may use the following theorem:

Theorem 5.14. Suppose p ≥ 3, q ≥ 1. Let Λ ∈ Ln be a unimodular latticesuch that HΛ is not closed. Let ν be any continuous function on K. Then

(34) limt→+∞

∫K

f̃(atkΛ)ν(k) dm(k) =

∫K

ν dm

∫Ln

f̃(∆) dµ(∆).

To prove Theorem 5.6 we use the following generalization:

Theorem 5.15. Suppose p ≥ 3, q ≥ 1. Let ν be as in Theorem 5.14, and letC be any compact set in Ln. Then for any ϵ > 0 there exist finitely many pointsΛ1, . . . ,Λℓ ∈ Ln such that

(i) The orbits HΛ1, . . . , HΛℓ are closed and have finite H-invariant measure.(ii) For any compact subset F of C \

∪1≤i≤ℓHΛi, there exists t0 > 0, so that

for all Λ ∈ F and t > t0,

(35)

∣∣∣∣∫K

f̃(atkΛ)ν(k) dm(k)−∫Lnf̃ dµ

∫K

ν dm

∣∣∣∣ ≤ ϵTheorem 5.14 and Theorem 5.15 if f̃ is replaced by a bounded function ϕ.If we replace f̃ by a bounded continuous function ϕ then (34) and (35) follow easilyfrom Theorem 4.4. (This was the original motivation for Theorem 4.4). The factthat Theorem 4.4 deals with unipotents and Theorem 5.15 deals with large spheresis not a serious obstacle, since large spheres can be approximated by unipotents.In fact, the integral in (34) can be rewritten as∫

B

(1

T (x)

∫ T (x)0

ϕ(utx) dm(k)

)dx,

where B is a suitable subset of G and U is a suitable unipotent. Now by Theo-rem 4.4, the inner integral tends to

∫G/Γ

ϕ uniformly as long as x is in a compact set

away from an explicitly described set E, where E is a finite union of neighborhoodsof sets of the form π(C) where C is a compact subset of some N(F,U). By directcalculation one can show that only a small part of B is near E, hence Theorem 5.14and Theorem 5.15 both hold.

Remark. Both Theorem 4.4 and Ratner’s uniform distribution theorem Theo-rem 1.12 hold for bounded continuous functions, but not for arbitrary continuousfunctions from L1(G/Γ). However, for a non-negative bounded continuous function


f on Rn, the function f̃ defined in (30) is non-negative, continuous, and L1 but un-bounded (it is in Ls(G/Γ) for 1 ≤ s < n, where G = SL(n,R), and Γ = SL(n,Z)).The lower bounds. As it was done in [DM4] it is possible to obtain asymp-

totically exact lower bounds by considering bounded continuous functions ϕ ≤ f̃ .However, to prove the upper bounds in the theorems stated above we need to exam-ine carefully the situation at the “cusp” of G/Γ, i.e outside of compact sets. Thiswill be done in §6.

6. Quantitative Oppenheim (upper bounds)

The references for this section are [EMM1] and [EMM2].

Lattices. Let ∆ be a lattice in Rn. We say that a subspace L of Rn is ∆-rational ifL∩∆ is a lattice in L. For any ∆-rational subspace L, we denote by d∆(L) or simplyby d(L) the volume of L/(L ∩∆). In the notation of [Kl1, §3], d∆(L) = ∥L ∩∆∥.

Let us note that d(L) is equal to the norm of e1 ∧ · · · ∧ eℓ in the exterior power∧ℓ(Rn) where ℓ = dimL and (e1, · · · , eℓ) is a basis over Z of L∩∆. If L = {0} we

write d(L) = 1.Let us introduce the following notation:

αi(∆) = sup{ 1d(L)

∣∣∣ L is a ∆-rational subspace of dimension i }, 0 ≤ i ≤ n,α(∆) = max

0≤i≤nαi(∆).

(36)

The following lemma is known as the “Lipshitz Principle”:

Lemma 6.1 ([Sch, Lemma 2]). Let f be a bounded function on Rn vanishingoutside a compact subset. Then there exists a positive constant c = c(f) such that

f̃(∆) < cα(∆)

for any lattice ∆ in Rn. Here f̃ is the function on the space of lattices defined in(30).

Replacing f̃ by α. By Lemma 6.1, the function f̃(g) on the space of unimodularlattices Ln is majorized by the function α(g). The function α is more convenientsince it is invariant under the left action of the maximal compact subgroup K̂ ofG, and its growth rate at infinity is known explicitly. Theorems 5.2 and 5.6 areproved by combining Theorem 4.4 with the following integrability estimate:

Theorem 6.2 ([EMM1]). If p ≥ 3, q ≥ 1 and 0 < s < 2, or if p = 2, q ≥ 1and 0 < s < 1, then for any lattice ∆ in Rn

supt>0

∫K

α(atk∆)s dm(k)

30 ALEX ESKIN

that in the case q = 1, the rank of X is 1, and the sets KatK are metric spheres ofradius t, centered at the origin.

If (p, q) = (2, 1) or (2, 2), Theorem 6.2 does not hold even for s = 1. Thefollowing result is, in general, best possible:

Theorem 6.3 ([EMM1]). If p = 2 and q = 2, or if p = 2 and q = 1, then forany lattice ∆ in Rn,

(37) supt>1

1

t

∫K

α(atk∆) dm(k) r}. Choose a continuous nonnegativefunction gr on G/Γ such that gr(x) = 1 if x ∈ A(r + 1), gr(x) = 0 if x /∈ A(r) and0 ≤ gr(x) ≤ 1 if x ∈ A(r)−A(r + 1). Then∫

K

f̃(atkx)ν(k) dm(k) =

=

∫K

(f̃gr)(atkx)ν(k) dm(k) +

∫K

(f̃ − f̃gr)(atkx)ν(k) dm(k).(38)

But (letting β = 2 − s), (f̃gr)(y) ≤ B1α(y)2−βgr(y) = B1α(y)2−β2 gr(y)α(y)

− β2 ≤B1r

− β2 α(y)2−β2 (the last inequality is true because gr(y) = 0 if α(y) ≤ r). Therefore

(39)

∫K

(f̃gr)(atkx)ν(k) dm(k) ≤ B1r−β2

∫K

α(atkx)2− β2 ν(k) dm(k).

According to Theorem 6.2 there exists B such that∫K

α(atkx)2− β2 dm(k) < B

for any t ≥ 0 and uniformly over x ∈ C. Then (39) implies that

(40)

∫K

(f̃gr)(atkx)ν(k) dm(k) ≤ BB1(sup ν)r−β2 .

Since the function f̃ − f̃gr is continuous and has a compact support, the “boundedfunction” case of Theorem 5.15 implies that for every ϵ > 0 there exists a finite setof points x1, . . . , xℓ with Hxi closed for each i so that for every compact subset F

of C \∪ℓ

i=1Hxi there exists t0 > 0 such that for every t > t0 and every x ∈ F ,(41)∣∣∣∣∣∫K

(f̃ − f̃gr)(atkx)ν(k) dm(k)−∫G/Γ

(f̃ − f̃gr)(y) dµ(y)∫K

ν(k) dm(k)

∣∣∣∣∣ < ϵ2 .It is easy to see that (38), (40) and (41) imply (35) if r is sufficiently large. Thisimplies Theorem 5.15. �

In the rest of this section, we prove Theorem 6.2 and Theorem 6.3. We recallthe notation from §5: G is SL(n,R), Γ = SL(n,Z), K̂ ∼= SO(n) is a maximalcompact subgroup of G, H ∼= SO(p, q) ⊂ G, K = H ∩ K̂ is a maximal compactsubgroup of H, and X is the symmetric space K\H. From its definition (36), thefunction α(∆) is the maximum over 1 ≤ i ≤ n of K̂ invariant functions αi(∆). The


main idea of the proof is to show that the αsi satisfy a certain system of integralinequalities which imply the desired bounds.

If p ≥ 3 and 0 < s < 2, or if (p, q) = (2, 1) or (2, 2) and 0 < s < 1, we showthat for any c > 0 there exist t > 0, and ω > 1 so that the the functions αsi satisfythe following system of integral inequalities in the space of lattices:

(42) Atαsi ≤ ciαsi + ω2 max

0

32 ALEX ESKIN

Lemma 6.5. Let H ∼= SO(2, 1) be the orthogonal group of the quadratic formx2 + y2 − z2. Let {at | t ∈ R} be a self-adjoint one-parameter subgroup of H, andlet K = H∩O(3) denote the maximal compact of H. We define another norm ∥·∥∗on R3 by

(46) ∥(x, y, z)∥∗ = max(√x2 + y2, |z|).

Then, for any v ∈ R3, v ̸= 0, and any t > 0,

(47)

∫K

dm(k)

∥atkv∥∗≤ 1

∥v∥∗.

6.2. A system of inequalities.

Lemma 6.6. For any two ∆-rational subspaces L and M

(48) d(L)d(M) ≥ d(L ∩M)d(L+M).

Proof. Let π : Rn → Rn/(L ∩M) denote the natural projection. Then d(L) =d(π(L))d(L∩M), d(M) = d(π(M))d(L∩M) and d(L+M) = d(π(L+M))d(L∩M).On the other hand the inequality (48) is equivalent to the inequality

d(L)

d(L ∩M)d(M)

d(L ∩M)≥ d(L+M)d(L ∩M)

.

Therefore replacing L,M and L+M by π(L), π(M) and π(L+M) we can assumethat L∩M = {0}. Let (e1, · · · , eℓ), ℓ = dimL, and (eℓ+1, · · · , eℓ+m), m = dimM ,be bases in L and M respectively. Then

(49) d(L)d(M) = ∥e1 ∧ · · · ∧ eℓ∥ ∥eℓ+1 ∧ · · · ∧ eℓ+m∥≥ ∥e1 ∧ · · · ∧ eℓ ∧ eℓ+1 ∧ · · · ∧ eℓ+m∥ ≥ d(L+M)

that proves (48) (the second inequality in (49) is true because (L∩∆)+(M ∩∆) ⊂(L+M) ∩∆. �

Lemma 6.7. Let {at | t ∈ R} be a self-adjoint one-parameter subgroup ofSO(2, 1). Let p and q be positive integers, and denote p+ q by n. Denote SO(p)×SO(q) by K. Suppose p ≥ 3, q ≥ 1 and 0 < i < n, or p = 2, q = 2 and i = 1 or 3.Then for any s, 0 < s < 2, and any c > 0 there exist t > 0 and ω > 1 such that forany lattice Λ in Rn

(50)

∫K

αi(atkΛ)s dm(k) <

c

2αi(Λ)

s + ω2 max0 0 there exist t > 0 and ω > 1 such that (50) holds.

Proof. Fix c > 0. In view of Proposition 6.4 one can find t > 0 such that∫K

dm(k)

∥atkv∥s<c

2,

whenever v ∈ F (i), ∥v∥ = 1. It follows that

(51)

∫K

dm(k)

∥atkv∥s<c

2· 1∥v∥s

,


for any v ∈ F (i), v ̸= 0. Let Λ be a lattice in Rn. There exists a Λ-rational subspaceLi of dimension i such that

(52)1

dΛ(Li)= αi(Λ).

The inequality (51) implies

(53)

∫K

dm(k)

datkΛ(atkLi)s<c

2

1

dΛ(Li)s.

Let ω = max00. Now using (52), (54) and Lemma 6.6 we get that for any k ∈ K

αi(atkΛ) < ωαi(Λ) =ω

dΛ(Li)<

ω2√dΛ(Li)dΛ(M)

≤ ω2√

dΛ(Li ∩M)dΛ(Li +M)

≤ ω2√αi+j(Λ)αi−j(Λ).

(57)

Hence if Ψi ̸= {Li}

(58)

∫K

αi(atkΛ)s dm(k) ≤ ω2 max

0

34 ALEX ESKIN

Lemma 6.8. Let {at | t ∈ R} be a self-adjoint one-parameter subgroup of H =SO(2, 1), and denote SO(2) by K. Then there exist t0 > 0 and ω > 1, such thatfor any t < t0, for any unimodular lattice Λ in R3, and 1 ≤ i ≤ 2,

(61)

∫K

α∗i (atkΛ) dm(k) < α∗i (Λ) + ω

2√α3−i(Λ).

Proof. The argument is identical to the proof of Lemma 6.7 except that one usesLemma 6.5 instead of Proposition 6.4. �

Now let H = SO(2, 2). The space V =∧2

(R4) splits as a direct sum V1 ⊕ V2of two invariant subspaces, where on each Vi, H preserves a quadratic form Qi ofsignature (2, 1). We define on each Vi a Euclidean norm ∥ · ∥∗i by (46) (adapted toQi). Let πi denote the orthogonal projections from V to Vi. Now let ∆ be a latticein R4, and let L be a two-dimensional ∆-rational subspace of R4. For 1 ≤ i ≤ 2,let

(62) di,#∆ (L) = ∥πi(e1 ∧ e2)∥∗i ,

where {e1, e2} is a basis over Z for ∆ ∩ L. Then let

(63) α#2 (∆) = supL

{min

(1

d1,#∆ (L),

1

d2,#∆ (L)

)}.

The supremum is taken over ∆-rational two dimensional subspaces L. By construc-tion, for any ∆,

(64) C−1α#2 (∆) < α2(∆) < Cα#2 (∆),

where C is an absolute constant.

Lemma 6.9. Let {at | t ∈ R} be a self-adjoint one-parameter subgroup ofSO(2, 1), where SO(2, 1) is diagonally embedded in H = SO(2, 2), under its lo-cal identification with SL(2,R)× SL(2,R). Denote SO(2)× SO(2) by K, and themaximal compact of SO(2, 1) by K̃. Then there exist t0 > 0 and ω > 1, such thatfor any t < t0 and for any unimodular lattice Λ in R4,

(65)

∫K̃

α#2 (atk̃Λ) dm(k̃) < α#2 (Λ) + ω

2√α1(Λ)α3(Λ).

Proof. The group K̃ is diagonally embedded in K. Recall that each SO(2, 2)

invariant subspace Vi of∧2

(R4) is fixed pointwise by one of the SL(2,R) factors,while the other fixes a quadratic form of signature (2, 1). Thus, for 1 ≤ i ≤ 2, theinequalities:

(66)

∫K̃

dm(k̃)

∥πi(atk̃v)∥∗i≤ 1

∥πi(v)∥∗i


follow immediately from Lemma 6.5. Hence,∫K̃

min

(1

∥π1(atk̃v)∥∗1,

1

∥π2(atk̃v)∥∗2

)dm(k)

≤ min

(∫K̃

dm(k̃)

∥π1(atk̃v)∥∗1,

∫K̃

dm(k̃)

∥π2(atk̃v)∥∗2

)

≤ min(

1

∥π1(v)∥∗1,

1

∥π2(v)∥∗2

).(67)

The rest of the proof is identical to that of Lemma 6.7 except that (67) is used inplace of Proposition 6.4. �

6.3. Coarsely Superharmonic Functions. Let n ∈ N+ and let D+n de-note the set of diagonal matrices d(λ1, · · · , λn) ∈ GL(n,R) with λ1 ≥ λ2 ≥· · · ≥ λn > 0. For any g ∈ GL(n,R), consider the Cartan decomposition g =k1(g)d(g)k2(g), k1(g), k2(g) ∈ K = O(n,R), d(g) ∈ D+n and denote by λ1(g) ≥λ2(g) ≥ · · · ≥ λn(g) the eigenvalues of d(g).

Lemma 6.10. For every ϵ > 0 there exists a neighborhood U of e in O(n,R)such that

(68)∣∣∣ λi(d1kd2)λi(d1)λi(d2)

�

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